The short answer is that you can burnish the flesh side for a nice smooth surface. If you want color the flesh side of the bag. To get a very nice, smooth finish on the flesh side, spread a light coat of gum tragacanth on a section of the flesh side and let it dry a bit, then take a bone creaser and rub down the section you covered. Repeat the process until you've covered the entire grain side. This process will partially seal the grain side and smooth it out.
Like you said, another option is to line it with pigskin or another light lining leather.
Hope this helps.
Ric